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Can You Solve the Million-Dollar, Unsolvable Chess Problem?

The queen of all puzzles.

The problem may not be intrinsically challenging by nature, but increasing the size of the board pushes it from complex to nigh impossible.
The problem may not be intrinsically challenging by nature, but increasing the size of the board pushes it from complex to nigh impossible. Public Domain

FACED WITH SEEMINGLY UNSOLVABLE PROBLEMS, historically, people get creative, whether a sword through the Gordian Knot or the threat of one through a disputed baby. But a seemingly “simple” chess problem will require a sharper solution—so sharp, in fact, that researchers at the University of St. Andrews in Scotland believe it could earn its master one of the $1 million Millennium Prizes, from the Clay Mathematics Institute.

The riddle is based on what is known as the Queens Puzzle, first devised in 1850. Eight queens must be placed on a standard chessboard so that no two pieces can take one another. According to a release from the university, “This means putting one queen each row, so that no two queens are in the same column, and no two queens in the same diagonal.” Solutions are not hard to imagine, but the problem becomes more complex when the chessboard grows—say 100 queens on a 100-by-100 chessboard.

New research from computer science professors Ian P. Gent, Christopher Jefferson, and Peter Nightingale refers to a still more challenging variant in which the board is even larger, but some queens have already been placed. In an interview with the Clay Mathematics Institute, Gent said this problem, technically known as the “n-Queens Completion Problem,” falls into a class of high-level math puzzles known as “NP-Complete.” Any algorithm that could solve it, Gent said, could therefore be used indirectly to solve others in the class—and be a contender for the Millennium Prize.

A program of this sort would be far more powerful than anything we currently have, said Gent. “If you could write a computer program that could solve the problem really fast, you could adapt it to solve many of the most important problems that affect us all daily.” This program, he said, would be able to decrypt even the toughest online security, something that would take current software thousands of years to unravel, by scrolling through and then discarding an almost infinite number of solutions until one works. Nightingale, his colleague, questioned whether this is even be possible. “What our research has shown is that—for all practical purposes—it can’t be done,” he said.

Although it’s hard to prove definitively, historians believe chess was invented in around the year 570, in what is now northeastern India. There is no shortage of famous chess puzzles, many of which remain unsolved to this day. A more recent development, however, has come in the writing of programs that create or solve problems too difficult or time-consuming for humans to do unassisted.

Some of these programs are so complicated that even their designers don’t fully understand how they work. Chesthetica, a program written by the computer scientist Azlan Iqbal, generates hundreds of problems, using digital synaptic neural substrate (DSNS) technology. “One might ask where does Chesthetica ‘get its ideas’?” writes Iqbal in Chess News. “I do not know. How or why should a computer be able to compose chess problems like these at all? Can computers autonomously do this sort of thing? These are also good questions and I believe the answer lies with the DSNS technology.” Why it works, he explained, remained an open question—but, somehow, it does. Maybe the large Queens puzzle–solving program will be similarly inscrutable.

This article has been updated to reflect the difference between the Queens Puzzle and the “n-Queens Completion Puzzle,” and the source of a potential $1 million prize, which is the Clay Mathematics Institute.